sensors

Review

Dissipation Analysis Methods and Q-EnhancementStrategies in Piezoelectric MEMS Laterally VibratingResonators: A Review

Cheng Tu 1 , Joshua E.-Y. Lee 2,3 and Xiao-Sheng Zhang 1,*1 School of Electronic Science and Engineering, University of Electronic Science and Technology of China,

Chengdu 611731, China; ctu@uestc.edu.cn2 Department of Electrical Engineering, City University of Hong Kong, Kowloon, Hong Kong;

josh.lee@cityu.edu.hk3 State Key Laboratory of Terahertz and Millimeter Waves, City University of Hong Kong,

Kowloon, Hong Kong* Correspondence: zhangxs@uestc.edu.cn

Received: 5 August 2020; Accepted: 1 September 2020; Published: 2 September 2020�����������������

Abstract: Over the last two decades, piezoelectric resonant sensors based on micro-electromechanicalsystems (MEMS) technologies have been extensively studied as such sensors offer several uniquebenefits, such as small form factor, high sensitivity, low noise performance and fabrication compatibilitywith mainstream integrated circuit technologies. One key challenge for piezoelectric MEMS resonantsensors is enhancing their quality factors (Qs) to improve the resolution of these resonant sensors.Apart from sensing applications, large values of Qs are also demanded when using piezoelectricMEMS resonators to build high-frequency oscillators and radio frequency (RF) filters due to the factthat high-Q MEMS resonators favor lowering close-to-carrier phase noise in oscillators and sharpeningroll-off characteristics in RF filters. Pursuant to boosting Q, it is essential to elucidate the dominantdissipation mechanisms that set the Q of the resonator. Based upon these insights on dissipation,Q-enhancement strategies can then be designed to target and suppress the identified dominant losses.This paper provides a comprehensive review of the substantial progress that has been made duringthe last two decades for dissipation analysis methods and Q-enhancement strategies of piezoelectricMEMS laterally vibrating resonators.

Keywords: micro-electromechanical systems (MEMS); piezoelectric MEMS resonators; laterallyvibrating resonators; quality factors; dissipation mechanisms; Q-enhancement strategies

1. Introduction

Resonant sensors, characterized by a resonant frequency dependent on the physical measurand,offer a great advantage over conventional analog sensors in that a frequency outputs can be digitizedthrough mature frequency counting techniques realizable in digital systems [1]. This greatly reducesthe complexity and cost generally associated with using the analog-to-digital converters to measurethe amplitude of analogue voltages. As the information of the measurand is carried by frequencyinstead of amplitude, resonant sensors usually exhibit better immunity to interference or noisecompared to their analog counterparts [2]. One of the widely-used resonant sensors is the quartzcrystal microbalance (QCM), the operation of which relies on direct and inverse piezoelectric effectsinherent to quartz resonators [3]. Although QCMs have high Q, large power handling capabilityand excellent temperature stability for certain cut angles, miniaturization of quartz resonators isstill very challenging, which makes it difficult to integrate the quartz resonators monolithically withintegrated circuits (IC) [4]. With the rapid development of micro-fabrication techniques, resonant

Sensors 2020, 20, 4978; doi:10.3390/s20174978 www.mdpi.com/journal/sensors

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sensors based on various micromachined piezoelectric resonators, such as thin film bulk acousticwave resonators (FBARs) [5–7], laterally vibrating resonators (LVRs) [8–11] and flexural-mode beamresonators [12–14], have been proposed with fabrication processes compatible with mainstream ICtechnologies. For FBARs and flexural-mode beam resonators, the sensitivity per unit area and resonantfrequency are both dependent on the thickness of resonant structures, which makes it hard to decouplethese two important design parameters [15]. Moreover, FBARs normally have resonant frequencies inthe GHz range due to difficulty in depositing thick piezoelectric films with good quality. It shouldbe noted that a higher operating frequency of a resonant sensor does not necessarily translate intoimproved overall sensor performance as the noise level usually deteriorates with frequency, and itis generally accepted that higher operating frequency results in larger power consumption anddesign complexity for the read-out electronic circuit [15]. In contrast to FBARs and flexural beamresonators, LVRs feature lithographically definable resonant frequencies that are independent ofthickness. Therefore, the sensitivity of resonant sensors based on LVRs can be improved by reducingthe thickness while keeping resonant frequencies unchanged, unlike FBARs. This unique characteristic,combined with compatible manufacturing process with ICs, has rendered LVR a viable candidate forfuture integrated resonant sensors with miniaturized dimensions, high performance and low powerconsumption. Over the last two decades, a significant number of high-performance resonant sensorsbased on LVRs have been demonstrated, such as chemical sensors [16,17], thermal detectors [9,18],infrared sensors [19], biological sensors [20,21], inertial sensors [22,23], magnetometers [24,25] andminiaturized acoustic antennas [26].

In piezoelectric MEMS LVRs, the electric field is mainly applied across the thickness of thepiezoelectric film to generate a lateral strain within the plane of the device with suspended free edges.The amplitude of lateral vibration becomes maximum when the frequency of the excitation signalcoincides with the mechanical resonance of the resonator. For further details on the working principle ofpiezoelectric LVRs, we refer readers to references [15,27]. There are two types of electrode configurationsthat are commonly adopted for LVRs. The first type uses interdigital transducer (IDT) electrodesonly on one side of piezoelectric film [28,29], while the second type has top and bottom electrodeson both sides of piezoelectric film [30,31]. Depending on the structure of resonators, piezoelectricMEMS LVRs can also be categorized into two types. The first type of LVRs consists of a piezoelectricfilm, both as the transducer and acoustic cavity [32]. The commonly-used piezoelectric materialsinclude aluminum nitride (AlN) [32], lithium niobate (LN) [28], zinc oxide (ZnO) [27], gallium nitride(GaN) [33] and lead zirconate titanate (PZT) [34]. The second type of LVRs uses the thin piezoelectricfilm as a transducer only while having most of the acoustic energy propagating in the underlyingthicker substrate [27]. Such resonators are usually referred to as thin-film piezoelectric-on-substrate(TPoS) resonators. Compared to the LVRs using only the piezoelectric film as the resonator body,TPoS LVRs generally exhibit higher Q and larger power handling capability when appropriate substratematerials are adopted. The commonly-used substrate materials include silicon [35], diamond [36]and silicon carbide [37]. As a trade-off, the effective electromechanical coupling (keff

2) of TPoS LVRs isusually lower than piezoelectric-only LVRs as a significant part of the electrical energy transformedinto acoustic energy is stored in the substrate layer. To improve the keff

2 of TPoS, a novel type ofsidewall-excited LVRs has been reported as a potential solution [38].

Figure 1a shows a typical structure of a TPoS resonator, which consists of input/output IDTelectrodes, piezoelectric film, bottom electrode and substrate layer. When Si is used as the substratelayer, a bottom electrode layer could be omitted by highly doping the top surface of Si substrateand let it serve as the “bottom electrode” [39]. The applied AC electric field across the thicknessof the piezoelectric film (denoted as z direction) causes it to expand and contract in the y directionthrough the piezoelectric coefficient d31. Figure 1b,c depicts the perspective and cross-sectional viewsof the indented fundamental symmetrical (S0) mode of Lamb wave. It should be noted that theelectromechanical transduction efficiency is maximized when the electric field and strain field in thepiezoelectric film are matched. That is to say, the indented S0 mode can be efficiently excited when the

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center-to-center pitch of the electrodes equals to half wavelength (λ) of the Lamb wave as shown inFigure 1c. As such, the resonant frequency of the LVR is given by:

f0 =Vphase

λ≈

12Wpitch

√Es

ρs(1)

where Vphase denotes the acoustic phase velocity of the resonator. Es and ρs are the Young’s modulus andmass density of the substrate layer, respectively. It should be noted that Equation (1) approximates theacoustic phase velocity using only the material properties of the substrate layer, which is usually muchthicker than the other constituent layers of TPoS LVRs. Equation (1) indicates a key feature of LVRs,which is that the resonant frequency of the device can be set by the lithographically defined dimension.Although Equation (1) uses variable “Wpitch” for the case where IDT electrodes are adopted, one cansimply replace Wpitch with other physical dimensions that represent half of acoustic wavelength in LVRsusing patch electrodes (e.g., the width of the resonator in the width-extensional LVRs [39]). Figure 1dshows the electrical characterization setup for a 2-port LVR, from which the network parameters suchas Z, Y and S parameters can be obtained. A typical plot of the associated transfer admittance Y21

is shown in Figure 1e, where it can be seen that the magnitude of Y21 is maximum at the resonantfrequency f 0.

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12

phase s0

pitch s

V EfWλ ρ

= ≈

(1)

where Vphase denotes the acoustic phase velocity of the resonator. Es and ρs are the Young’s modulus and mass density of the substrate layer, respectively. It should be noted that Equation (1) approximates the acoustic phase velocity using only the material properties of the substrate layer, which is usually much thicker than the other constituent layers of TPoS LVRs. Equation (1) indicates a key feature of LVRs, which is that the resonant frequency of the device can be set by the lithographically defined dimension. Although Equation (1) uses variable “Wpitch” for the case where IDT electrodes are adopted, one can simply replace Wpitch with other physical dimensions that represent half of acoustic wavelength in LVRs using patch electrodes (e.g., the width of the resonator in the width-extensional LVRs [39]). Figure 1d shows the electrical characterization setup for a 2-port LVR, from which the network parameters such as Z, Y and S parameters can be obtained. A typical plot of the associated transfer admittance Y21 is shown in Figure 1e, where it can be seen that the magnitude of Y21 is maximum at the resonant frequency f0.

Figure 1. (a) Schematic for typical structure of thin-film piezoelectric-on-substrate (TPoS) resonators; (b,c) perspective and cross-sectional views of the vibration mode shape of 7th-order fundamental symmetrical (S0) mode associated with resonant frequency f0; (d) electrical characterization setup for a 2-port TPoS laterally vibrating resonators (LVR); (e) typical measured magnitude of transfer admittance Y21 of a 2-port TPoS LVR.

The operation of resonant sensors hinges on the shift in resonant frequency due to a change in effective mass or stiffness of the resonant structure when subjected to a perturbation from the quantity of interest. There are three methods that are commonly used to detect the frequency shift of LVRs. The first utilizes the electrical characterization setup shown in Figure 1d, where the frequency response of the resonator is monitored by a network analyzer [10]. The second method involves detecting a change in amplitude of the output signal by exciting the resonator with an AC signal at a fixed frequency around the resonant frequency. This method essentially converts the shift in resonant frequency to a change in amplitude which is usually monitored using a lock-in amplifier [40]. The third method involves building an oscillator by connecting the resonator with the electronic amplifier to form a feedback loop. The output frequency of the oscillator can be readily monitored by a frequency counter [24]. Compared to the first two methods, the third method is more suitable for commercial applications as it does not require cumbersome characterization equipment [41]. It should be noted that, in resonant sensors characterized by all three methods mentioned above, higher Q is generally desired for better resolution [4]. Thus, it is of vital importance to maximize the Q of LVRs for sensing applications. In addition, boosting Q also lowers close-to-carrier phase noise in the application of oscillators [42], and improves roll-off characteristics in the application of RF filters [43].

This paper is organized as follows: Section 2 of this paper introduces different dissipation sources in piezoelectric LVRs, with particular attention on anchor loss and thermoelastic damping

Figure 1. (a) Schematic for typical structure of thin-film piezoelectric-on-substrate (TPoS) resonators;(b,c) perspective and cross-sectional views of the vibration mode shape of 7th-order fundamentalsymmetrical (S0) mode associated with resonant frequency f 0; (d) electrical characterization setup for a2-port TPoS laterally vibrating resonators (LVR); (e) typical measured magnitude of transfer admittanceY21 of a 2-port TPoS LVR.

The operation of resonant sensors hinges on the shift in resonant frequency due to a change ineffective mass or stiffness of the resonant structure when subjected to a perturbation from the quantityof interest. There are three methods that are commonly used to detect the frequency shift of LVRs.The first utilizes the electrical characterization setup shown in Figure 1d, where the frequency responseof the resonator is monitored by a network analyzer [10]. The second method involves detectinga change in amplitude of the output signal by exciting the resonator with an AC signal at a fixedfrequency around the resonant frequency. This method essentially converts the shift in resonantfrequency to a change in amplitude which is usually monitored using a lock-in amplifier [40]. The thirdmethod involves building an oscillator by connecting the resonator with the electronic amplifier toform a feedback loop. The output frequency of the oscillator can be readily monitored by a frequencycounter [24]. Compared to the first two methods, the third method is more suitable for commercialapplications as it does not require cumbersome characterization equipment [41]. It should be notedthat, in resonant sensors characterized by all three methods mentioned above, higher Q is generallydesired for better resolution [4]. Thus, it is of vital importance to maximize the Q of LVRs for sensing

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applications. In addition, boosting Q also lowers close-to-carrier phase noise in the application ofoscillators [42], and improves roll-off characteristics in the application of RF filters [43].

This paper is organized as follows: Section 2 of this paper introduces different dissipation sourcesin piezoelectric LVRs, with particular attention on anchor loss and thermoelastic damping (TED),both of which are generally accepted to be dominant in setting Q of LVRs. Section 3 reviews therecent progress on numerical and experimental methods for analyzing these two dominant dissipationmechanisms. In Section 4, a variety of Q-enhancement strategies for reducing anchor loss or TED arepresented and compared. Section 5 discusses the issues that need to be addressed in relation to existingdissipation analysis methods and Q-enhancement strategies.

2. Dominant Dissipations in LVRs

There are many dissipation mechanisms reported for LVRs, which are usually divided into twogroups, namely intrinsic and extrinsic loss. Intrinsic losses are fundamental dissipative processes,thus dependent on the material properties, mode of resonance and internal structure. In contrast,extrinsic losses stem from interactions between the structure and the environment. Generally,the intrinsic losses, such as phonon-phonon or phonon-electron interactions, set the maximum Qachievable in bulk mode resonators [44]. Thermoelastic damping (TED) is one example of a type ofintrinsic loss [45]. In most of the cases, the intrinsic losses are much smaller compared to extrinsiclosses and easily get masked by the latter. Extrinsic losses mainly include anchor loss [46], ohmic loss,viscous loss [47] and surface loss [48], each of which will be detailed in the following part of this paper.

Quality factor (Q) is defined as the ratio of the maximum energy stored in the resonant systemover the energy dissipated per cycle. The overall quality factor (Qtotal) of LVRs can be expressed bysumming up the distinct dissipations [49]:

Qtotal = 2πEstored

Edissipated= (

1Qanchor

+1

QTED+

1QOther

)−1

(2)

where Qanchor, QTED and Qother correspond to anchor loss, TED, and other losses, respectively.Unfortunately, there is scarcely any reported work on measuring individual loss (or individualQ) directly. Instead, only Qtotal, which describes the combined effect of different damping sources,can be obtained by measuring the amplitude-frequency or phase-frequency responses of the resonators.This brings about a huge challenge in analyzing the damping mechanisms present in LVRs becauseeach constituent source of damping cannot be easily analyzed separately. However, if the dominantlosses are much larger than other losses, then the other losses may be neglected in an approximation ofQtotal. In other words, the approximation considers only the dominant losses, which exhibit lower Qsand thus have more significant impact on Qtotal as indicated by Equation (2). It should be noted thatEquation (2) emphasizes Qanchor and QTED because anchor loss and TED were found to play dominantroles in setting Qtotal in many LVRs [46,50–52]. Thus, this work mainly focuses on these two dissipationsources. However, other dissipations could become dominant in certain cases, which will be detailedin the following sections.

2.1. Anchor Loss

Generally, LVRs consist of free-standing resonant bodies that require mechanical support (dubbedas anchor). Thus, an anchor is defined as a mechanical structure that attaches the resonator bodyto the peripheral supporting frame. Anchor loss occurs when elastic energy propagates from theresonator body to the surrounding substrate through the anchors. If the total elastic energy stored inthe resonator is given by Er and the energy propagating out via the anchors is given by El, then Qanchorcan be computed using the following equation [53]:

Qanchor = 2πEr

El(3)

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For any LVR with a free-standing resonator body which is suspended by the supporting structures,anchor loss is unavoidable. This is true even if the anchor structure is placed at the positions whereminimum displacement occurs (such points are normally referred to as nodal points). This is becausethe anchor structures have a finite size in practical cases rather than being an idealized infinitesimalpoint. It is widely reported that anchor loss plays the major role in setting the Q in most of LVRs [31,46].As such, a variety of Q-enhancement strategies that aim to reduce anchor loss have been proposedover the last two decades, some of which will be reviewed and discussed in detail in Section 4.

2.2. Electrode-Related Loss and Thermoelastic Damping (TED)

For LVRs utilizing piezoelectric transduction, metal electrodes are necessary for actuation anddetection of the Lamb wave in the resonator body. It was found there exists a damping mechanismrelated to the electrodes that has been associated with interfacial loss [54]. This electrode-relatedloss was confirmed by a demonstration in a 3.2-GHz overmoded LVR that the f ·Q product can bepushed to 1.17 × 1013 by reducing the electrodes coverage to 0.57% [55]. To completely eliminatethe electrode-related loss, some researchers proposed a capacitive-piezo transducer, which separatesthe electrodes from the resonator body by sub-micro gaps [56,57]. Applying this design strategy to a940-MHz LVR leads to a 5× enhancement in Q, resulting an f ·Q product 4.72 × 1012. These experimentalresults suggest that anchor loss alone cannot account for the measured Qs in LVRs, and theelectrode-related loss also plays a significant role in setting Q. However, there is yet to be a soliduniversal explanation for the electrode-related loss observed in various types of LVRs. The complexityin analyzing electrode-related loss arises due to the uncertainty of the thermal properties andinterfacial conditions for constituent metal and piezoelectric thin films [31,54]. One possible causefor the electrode-related loss is TED, which has been extensively studied for micro-resonators usingelectrostatic transduction [58–62]. For micro-resonators using piezoelectric transduction, it has beenexperimentally verified that TED plays a dominant role in setting Qs of a 1-GHz AlN LVR [51]. TED wasalso analyzed in a series of 140-MHz AlN-on-Si LVRs using finite-element analysis [50], where it wasshown that TED made comparable contributions as anchor loss in setting the Qs of regular flat-edgeLVRs. Thus, it is reasonable to consider TED as one of the significant source for electrode-related loss.

TED refers to an irreversible energy transfer process as elastic energy turns into heat [63,64].The dissipation is caused by thermal gradients arising from the local volumetric change as the resonatorsvibrate. As such, this dissipation mechanism is absent for pure shear modes, where the volume of suchmodes does not change. This is the reason why reported f ·Q products for Lamé bulk-mode siliconresonators can reach close to the limit set by intrinsic material loss [65]. In comparison, TED becomesmore relevant for Lamb wave modes where longitudinal waves and shear waves are usually coupledin the resonator body. It is reported that QTED can be evaluated using the following equation [63]:

QTED =cv

2

ψα2ρkTω(4)

where cv denotes heat capacity per unit volume, ψ a constant related to the geometry of the resonator,α thermal expansion coefficient, ρ mass density, k thermal conductivity, T absolute temperature andω angular frequency. Although Equation (4) holds for single-material devices, it can be extended topredict QTED of composite resonators (e.g., TPoS LVRs) by considering TED from each constituent layerwith the help of finite-element analysis tools [50]. It was also found that TED from metals far exceedsthat from semiconductors and dielectrics since metals generally exhibit larger thermal conductivitiesand thermal expansion coefficients [51,66]. Thus, it is reasonable to attribute TED to metal electrodesand simply ignore TED from piezoelectric and substrate layers as a reasonable approximation of Qtotal.

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2.3. Other Dissipation Sources

Apart from anchor loss and TED, other dissipations such as ohmic loss and viscous loss couldalso become dominant for LVRs in certain cases.

Ohmic loss occurs when electrical currents pass through electrodes with low but non-zero resistance.As the operation frequency of LVRs increases, ohmic losses usually become more significant due to thesmaller electrode width required for excitation of high-frequency Lamb waves. Larger electrical lossresults in more heat generation causing non-linearity issues in LVRs which further limits the powerhanding capability [67]. To reduce ohmic loss, a common method is to increase the thickness the metalelectrodes. However, a thick metal layer could negatively impact the indented vibration mode or causeadditional damping [51]. An alternative solution is to use low-resistivity metals. It should be notedthat trade-offs usually exist for choosing the metal materials for LVRs as many other design factors,such as acoustic impedance, power durability and complexity in terms of microfabrication process,also need to be taken into account.

Viscous loss occurs when a portion of kinetic energy transfers from the resonator to the surroundingmolecules. It usually becomes larger as the surface to volume ratio of the resonator increases [47].When operating LVRs in a fluidic medium, which is usually the case for biological sensing applications,viscous loss becomes the most dominant damping mechanism [21]. A common method to reduceviscous loss is operating the resonator in a shear mode as the shear acoustic wave does not displace themolecules perpendicular to the resonator surface [68].

Another dissipation mechanism that also heavily depends on the surface to volume ratio of theresonator body is surface loss. Surface losses are usually associated with the surface non-idealitiessuch as surface roughness, lattice defects, adsorbents and contamination, which could be criticalto the performance of resonators when the size of devices shrinks to nano-scale [69,70]. Althoughthe physics of surface losses are complicated, there has been an impressive effort to analyze thisdamping mechanism using a unified model [48]. It has also been reported that vacuum annealingcan be used to significantly reduce surface losses [71,72]. Given the growing research interest innano-electromechanical LVRs in recent years [11,19,26,73,74], it can be envisioned that more attentionwill be drawn to study surface losses in LVRs with nano-scale dimensions.

It has been reported that the damping mechanisms related to phonon-phonon and phonon-electroninteractions limit the Q of bulk (or contour) mode resonators [44,75]. But these losses have been shownto be negligible in LVRs as the experimentally demonstrated f ·Q products are at least an order ofmagnitude smaller than the theoretically predicted limit [55,76]. In addition, it has been proposed thatcharge redistribution loss could set the Q limit in piezoelectric resonators that operate at frequenciesless than 1 GHz [77]. However, there is a scarcity of published material regarding this loss mechanism.As such, these damping mechanisms are not covered in this review.

3. Dissipation Analysis Methods for LVRs

As anchor loss and TED are the most dominant loss mechanisms for LVRs, several methods ofanalysis targeting these two losses have been proposed during the last decade. These methods can bemainly categorized into two types, namely numerical analysis and experimental investigation methods,which can complement each other. This section will provide a comprehensive review on these twotypes of methods for analyzing anchor loss and TED.

3.1. Numerical and Experimental Analysis Methods for Anchor Loss

Anchor loss captures the radiation of elastic energy from the resonator body to surroundingsubstrate via the supporting structures. The analysis complexity lies in that a portion of elastic wavescould be reflected back into the resonator body if the surrounding substrate is finite in size or there existsan acoustic impedance mismatch along the path of outward radiation. In such case, one needs to take thewhole substrate in which resonator resides into account, which substantially increases the complexity

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of analysis. One commonly employed approach to this problem has been to consider the substrate as asemi-infinite domain where all outgoing elastic waves from the resonators are absorbed and no reflectionoccurs [78]. This approximation is reasonable considering the resonators are usually much smallercompared to the substrate. To mimic the semi-infinite substrate, perfectly matched layers (PMLs) wereusually adopted. The concept of PMLs was first introduced to absorb outgoing electromagnetic wavesto mimic infinitely large domains when solving problems of electromagnetic wave propagation [79].The ideal impedance matching characteristics of the PML is realized by coordinate transformationwhich force the displacements to vanish [80]. Thus, truncating the semi-infinite substrate with athree-dimensional PML with finite size allows the substrate to absorb the outgoing elastic waves fromany angle. Given that all the elastic energy arriving at the PML is completely dissipated, the PMLtechnique also facilitates computing the arriving elastic energy from the anchors and thus Qanchoraccording to Equation (3) [81].

As most of LVRs adopt straight beam tethers as the supporting structures connecting the resonatorbody with the substrate, it is meaningful to investigate the simplest case where a cantilever is attachedto the substrate. Frangi et al. [80] proposed the implementation of PML for a two-dimensional (2D)cantilever, as shown in Figure 2a. The length and width of the cantilever are L and H, respectively.As the cantilever vibrates, a portion of elastic wave propagates to the semi-infinite substrate on the leftcausing anchor loss. To mimic the semi-infinite substrate, a Cartesian PML with width of WPML is usedto truncate the substrate at certain distance (W∞) from the cantilever. The elastic waves radiate fromthe anchor of the cantilever to the substrate is completely dissipated in PML, which further allowscomputation of the elastic energy dissipated over one cycle. It should be noted that the computedelastic energy dissipated in PML heavily depends on the physical dimensions of PML such as W∞and WPML. Setting the value of WPML too small will cause a reflected wave that perturbs the solutionof the wave propagation in the substrate [78]. Specific criteria for the selection of PML parameters isdiscussed in detail in [80]. To evaluate the anchor loss more accurately, the implementation of PML forthe three-dimensional (3D) cantilever is necessary. Figure 2b shows the case where a 3D semi-infinitesubstrate is truncated by the PML. An alternative to implement PMLs is shown in Figure 2c, where asemi-infinite plate substrate with finite width (W) is truncated by the PML. In this case, besides W∞and WPML, it is also important to determine the width of the plate to obtain reliable predictions forQanchor. It can be seen from Figure 2d that the computed values of Qanchor tends to converge when W issufficiently large. This suggests there exists a minimum value for the plate width for reliable predictionsfor Qanchor. As the proposed numerical method is based on finite-element analysis, the reliability ofcomputed Qanchor needs to be confirmed by refining the mesh of the model, as shown in Figure 2e.

Based on the numerical analysis method using PML, Siddiqi et al. [50] proposed a 3D model toevaluate the anchor loss in a AlN-on-Si resonator (a typical type of TPoS LVR), as shown in Figure 3a.It can be seen from Figure 3a that, by utilizing the symmetry of the resonator, only a quarter sectionof the device is analyzed with the aim to reduce computation time. The physical dimensions of thesubstrate, PML and mesh element size should be set in relation to the acoustic wavelength (λ) of theLamb wave propagating in the resonator. Figure 3b shows the simulated vibration mode shape ofthe intended S0 mode, where the acoustic wavelength of 60 µm is labelled. The side length of thesquare substrate region is set to be about twice the acoustic wavelength, and the length of PML is set tobe equal to one acoustic wavelength. The convergence test for the computed Qanchor was performedover PML length and number of mesh elements, as shown in Figure 3c,d. Figure 3e shows a scanningelectron micrograph (SEM) for the tether part of the fabricated resonator. With the help of the numericalanalysis method using PML, the values of Qanchor were computed, which showed good agreement withthe experimental results. Furthermore, the proposed analysis method can be used to predict the Qanchoras functions of tether length, tether width and fillet radius as shown in Figure 3f–h. The work shownin [50] proves that the numerical analysis method based on PMLs is a powerful means to evaluateanchor loss which is expected to significantly benefit the design of high-Q LVRs.

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outgoing elastic waves from any angle. Given that all the elastic energy arriving at the PML is completely dissipated, the PML technique also facilitates computing the arriving elastic energy from the anchors and thus Qanchor according to Equation (3) [81].

As most of LVRs adopt straight beam tethers as the supporting structures connecting the resonator body with the substrate, it is meaningful to investigate the simplest case where a cantilever is attached to the substrate. Frangi et al. [80] proposed the implementation of PML for a two-dimensional (2D) cantilever, as shown in Figure 2a. The length and width of the cantilever are L and H, respectively. As the cantilever vibrates, a portion of elastic wave propagates to the semi-infinite substrate on the left causing anchor loss. To mimic the semi-infinite substrate, a Cartesian PML with width of WPML is used to truncate the substrate at certain distance (W∞) from the cantilever. The elastic waves radiate from the anchor of the cantilever to the substrate is completely dissipated in PML, which further allows computation of the elastic energy dissipated over one cycle. It should be noted that the computed elastic energy dissipated in PML heavily depends on the physical dimensions of PML such as W∞ and WPML. Setting the value of WPML too small will cause a reflected wave that perturbs the solution of the wave propagation in the substrate [78]. Specific criteria for the selection of PML parameters is discussed in detail in [80]. To evaluate the anchor loss more accurately, the implementation of PML for the three-dimensional (3D) cantilever is necessary. Figure 2b shows the case where a 3D semi-infinite substrate is truncated by the PML. An alternative to implement PMLs is shown in Figure 2c, where a semi-infinite plate substrate with finite width (W) is truncated by the PML. In this case, besides W∞ and WPML, it is also important to determine the width of the plate to obtain reliable predictions for Qanchor. It can be seen from Figure 2d that the computed values of Qanchor tends to converge when W is sufficiently large. This suggests there exists a minimum value for the plate width for reliable predictions for Qanchor. As the proposed numerical method is based on finite-element analysis, the reliability of computed Qanchor needs to be confirmed by refining the mesh of the model, as shown in Figure 2e.

Figure 2. Evaluation of anchor loss in a cantilever structure using numerical analysis: (a) implementation of perfectly matched layers (PML) for a 2D cantilever; (b) implementation of PML for a 3D cantilever with a semi-infinite substrate; (c) implementation of PML for a 3D cantilever with a plate substrate; (d) computed values of Qanchor; (e) mesh adopted for the 3D cantilever and semi-infinite substrate. This figure is reproduced from [80] (© 2013 Elsevier).

Figure 2. Evaluation of anchor loss in a cantilever structure using numerical analysis: (a) implementationof perfectly matched layers (PML) for a 2D cantilever; (b) implementation of PML for a 3D cantileverwith a semi-infinite substrate; (c) implementation of PML for a 3D cantilever with a plate substrate;(d) computed values of Qanchor; (e) mesh adopted for the 3D cantilever and semi-infinite substrate.This figure is reproduced from [80] (© 2013 Elsevier).

Damping in MEMS resonators are usually characterized by Q, which requires measurementof electrical parameters such as S, Z or Y parameters. The electrical characterization methods areconvenient and efficient in measuring Q, but the measured Q does not provide any insight of thenature of damping. Recently, Gibson et al. [82] investigated anchor loss using laser Doppler vibrometry(LDV). It is shown in the work that the undercutting region of a 220-MHz AlN LVR had an out-of-planedisplacement as large as 60 pm, which served as a proof that the device release distance had asignificant effect in determining the anchor loss. Tu et al. [83] also applied LDV technique to probethe anchor loss in AlN-on-Si LVRs, which showed a strong correlation between the measured Q andthe out-of-plane displacement profiles on the top surface of the resonator. In this work, two types ofresonator, namely flat-edge and biconvex resonators, were investigated by both LDV and electricalcharacterization methods. Figure 4a shows the SEM of a flat-edge resonator (F80). The flat-edgeresonator has a rectangular body with the interdigital electrodes placed on top, which is commonlyadopted configuration for LVRs. Figure 4b shows the SEM of the proposed biconvex resonator (B80),the edge of which is curved for confinement of acoustic energy into the center of the resonator and thusreduce anchor loss. The resonant frequency of the device is determined by the electrode pitch (Wp)which is equal to half of the acoustic wavelength (λ). Figure 4c depicts the schematic of a flat-edgeresonator showing the constituent layers of AlN-on-Si structure. The constituent layers are the same forboth flat-edge and biconvex resonators. The measured z-direction out-of-plane displacement profilesby LDV over four devices (F80, B80, F60 and B60) are shown in Figure 4d–g, respectively. It canbe seen from Figure 4d,f that maximum displacement with opposite signs occurs alternately alongthe width of the flat-edge resonators (F80 and F60). In the length direction, the displacement profileis generally uniform, which allows z-direction motions to be transferred to the supporting tethersresulting in anchor loss. In contrast, z-direction displacements are more concentrated in the center forresonators B80 and B60, as shown in Figure 4e–g. Intuitively, these measured results from LDV leads

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to a hypothesis that biconvex resonators suffer less anchor loss compared to conventional flat-edgeresonators since less acoustic energy is leaked to the surrounding substrate via supporting tethers in thelatter case. This hypothesis is verified by the measured magnitude of admittance frequency responsefrom the electrical characterization method as shown in Figure 4h, which shows that the measuredQ of resonator B60 is about 16 times that of F60. This strong correlation between the z-directiondisplacement profile and measured Q is consistently observed for resonators with different resonantfrequencies as shown in Figure 4i. The results shown in [83] validates that anchor loss can be probedby visualizing the out-of-plane displacement profile on the top surface of LVRs using LDV.

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Based on the numerical analysis method using PML, Siddiqi et al. [50] proposed a 3D model to evaluate the anchor loss in a AlN-on-Si resonator (a typical type of TPoS LVR), as shown in Figure 3a. It can be seen from Figure 3a that, by utilizing the symmetry of the resonator, only a quarter section of the device is analyzed with the aim to reduce computation time. The physical dimensions of the substrate, PML and mesh element size should be set in relation to the acoustic wavelength (λ) of the Lamb wave propagating in the resonator. Figure 3b shows the simulated vibration mode shape of the intended S0 mode, where the acoustic wavelength of 60 μm is labelled. The side length of the square substrate region is set to be about twice the acoustic wavelength, and the length of PML is set to be equal to one acoustic wavelength. The convergence test for the computed Qanchor was performed over PML length and number of mesh elements, as shown in Figure 3c,d. Figure 3e shows a scanning electron micrograph (SEM) for the tether part of the fabricated resonator. With the help of the numerical analysis method using PML, the values of Qanchor were computed, which showed good agreement with the experimental results. Furthermore, the proposed analysis method can be used to predict the Qanchor as functions of tether length, tether width and fillet radius as shown in Figure 3f–h. The work shown in [50] proves that the numerical analysis method based on PMLs is a powerful means to evaluate anchor loss which is expected to significantly benefit the design of high-Q LVRs.

Figure 3. Evaluation of anchor loss in a AlN-on-Si resonator using numerical analysis: (a) implementation of PML in a 3D model; (b) simulated vibration mode shape of S0 Lamb wave at 141.6 MHz; (c,d) computed Qanchor in relation to PML length and number of mesh elements; (e) scanning electron micrograph (SEM) for the tether part of the fabricated resonator; (f–h) computed Qanchor with respect to tether length, tether width and fillet radius. This figure is reproduced from [50] (© 2019 IOP Publishing).

Damping in MEMS resonators are usually characterized by Q, which requires measurement of electrical parameters such as S, Z or Y parameters. The electrical characterization methods are convenient and efficient in measuring Q, but the measured Q does not provide any insight of the nature of damping. Recently, Gibson et al. [82] investigated anchor loss using laser Doppler vibrometry (LDV). It is shown in the work that the undercutting region of a 220-MHz AlN LVR had

Figure 3. Evaluation of anchor loss in a AlN-on-Si resonator using numerical analysis:(a) implementation of PML in a 3D model; (b) simulated vibration mode shape of S0 Lamb wave at141.6 MHz; (c,d) computed Qanchor in relation to PML length and number of mesh elements; (e) scanningelectron micrograph (SEM) for the tether part of the fabricated resonator; (f–h) computed Qanchor withrespect to tether length, tether width and fillet radius. This figure is reproduced from [50] (© 2019IOP Publishing).

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an out-of-plane displacement as large as 60 pm, which served as a proof that the device release distance had a significant effect in determining the anchor loss. Tu et al. [83] also applied LDV technique to probe the anchor loss in AlN-on-Si LVRs, which showed a strong correlation between the measured Q and the out-of-plane displacement profiles on the top surface of the resonator. In this work, two types of resonator, namely flat-edge and biconvex resonators, were investigated by both LDV and electrical characterization methods. Figure 4a shows the SEM of a flat-edge resonator (F80). The flat-edge resonator has a rectangular body with the interdigital electrodes placed on top, which is commonly adopted configuration for LVRs. Figure 4b shows the SEM of the proposed biconvex resonator (B80), the edge of which is curved for confinement of acoustic energy into the center of the resonator and thus reduce anchor loss. The resonant frequency of the device is determined by the electrode pitch (Wp) which is equal to half of the acoustic wavelength (λ). Figure 4c depicts the schematic of a flat-edge resonator showing the constituent layers of AlN-on-Si structure. The constituent layers are the same for both flat-edge and biconvex resonators. The measured z-direction out-of-plane displacement profiles by LDV over four devices (F80, B80, F60 and B60) are shown in Figure 4d–g, respectively. It can be seen from Figure 4d,f that maximum displacement with opposite signs occurs alternately along the width of the flat-edge resonators (F80 and F60). In the length direction, the displacement profile is generally uniform, which allows z-direction motions to be transferred to the supporting tethers resulting in anchor loss. In contrast, z-direction displacements are more concentrated in the center for resonators B80 and B60, as shown in Figure 4e–g. Intuitively, these measured results from LDV leads to a hypothesis that biconvex resonators suffer less anchor loss compared to conventional flat-edge resonators since less acoustic energy is leaked to the surrounding substrate via supporting tethers in the latter case. This hypothesis is verified by the measured magnitude of admittance frequency response from the electrical characterization method as shown in Figure 4h, which shows that the measured Q of resonator B60 is about 16 times that of F60. This strong correlation between the z-direction displacement profile and measured Q is consistently observed for resonators with different resonant frequencies as shown in Figure 4i. The results shown in [83] validates that anchor loss can be probed by visualizing the out-of-plane displacement profile on the top surface of LVRs using LDV.

Figure 4. Experimental investigation of anchor loss using laser Doppler vibrometry: (a) scanning electron micrograph (SEM) of a flat-edge AlN-on-Si resonator; (b) SEM of a biconvex AlN-on-Si

Figure 4. Experimental investigation of anchor loss using laser Doppler vibrometry: (a) scanningelectron micrograph (SEM) of a flat-edge AlN-on-Si resonator; (b) SEM of a biconvex AlN-on-Si resonator;(c) schematic of a typical structure for AlN-on-Si LVR; (d–g) measured z-direction displacement profileon the top surface of four resonators (F80/B80 denotes flat-edge/biconvex resonator with acousticwavelength of 80µm); (h) measured magnitude of admittance frequency response for flat-edge resonator(F60) and biconvex resonator (B60), showing much higher Q of the latter; (i) comparison of measuredQs among three groups of device with different resonant frequencies (each group includes a flat-edgeresonator and a biconvex resonator, which were designed to have similar resonant frequencies butstarkly different Q). This figure is reproduced from [83] (© 2017 Elsevier).

3.2. Numerical and Experimental Analysis Methods for TED

The concept of TED was first proposed by Zener et al. in 1937 [84], where the thermoelasticcoupling between acoustic and thermal phonons was studied analytically for flexural vibration modesin thin rods. Although the Zener’s work aims at the fundamental mode of vibration in a beam structure,its implication with regards to the inverse relation between QTED and ambient temperature appliesgenerally for MEMS resonators of different shapes [85–89]. Much work has been done to improvethe accuracy of evaluating QTED in the context of MEMS resonators. Lifshitz et al. [90] proposed anexact analytical expression of QTED in thin rectangular beams undergoing small flexural vibrations.To accommodate more general cases other than simple beams, Ardito et al. [91] developed an ad-hocnumerical procedure which was implemented in a finite element program. Segovia-Fernandez etal. [51] proposed a semi-analytical approach that enables computation of QTED in LVRs by treating themetal electrodes as standard anelastic solids. The experimental results shown therein also validate thatTED in LVRs is mainly caused by metal electrodes.

Recently, Siddiqi et al. [50] applied Ardito’s formulation from [91] to a series of biconvex AlN-on-Siresonators which suffered more from TED rather than anchor loss. Figure 5a shows the computedthermal profile for a biconvex AlN-on-Si resonator under investigation, illustrating that the temperaturegradients occur alternatively along the width of the resonator. This temperature distribution patternagrees with the S0 vibration mode of Lamb wave indicating that these temperature variations arecaused by local compression and tension. Due to the presence of thermal gradients, irreversible heatflow arises resulting in TED. Figure 5b shows that the maximum changes in temperature occur at the

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electrode area which agrees with the analysis results shown in [51]. However, it is shown in [50] thatthe computed values of QTED from Al electrodes alone are about an order of magnitude larger thanthe experimental values of Q. Instead, by using the numerical analysis model, the computed QTED asa function of thermal expansion coefficients of AlN layer (αAlN) is obtained as shown in Figure 5d,from which it is suspected that the AlN layer has a greater contribution to TED compared to the Alelectrodes in AlN-on-Si resonators.Sensors 2020, 20, x FOR PEER REVIEW 11 of 25

Figure 5. Evaluation of thermoelastic damping (TED) using numerical analysis: (a) computed thermal profile showing temperature gradients as a result of local compression and tension; (b) computed thermal profile showing that maximum change in temperature occurs at electrodes; (c) computed QTED for both flat-edge and biconvex resonators as a function of temperature; (d) computed QTED as a function of thermal expansion coefficient of AlN layer (αAlN) for resonators with different curvatures (κ). This figure is reproduced from [50] (© 2019 IOP Publishing).

The key challenge to experimentally investigate TED in MEMS resonators is that TED is typically present together with anchor loss, which makes it difficult to analyze TED separately. Thus, it is important to find an approach to separate these two dissipation mechanisms. It is reported by Li et al. [92] that the measured Q was increased by about 2.5 times when operating a 61-MHz polysilicon wine-glass disk micro-resonators at cryogenic temperatures down to 5 K compared to room temperature. This experimental result suggests that at least one of the dominant dissipation mechanisms in silicon micro-resonators is dependent on temperature. This dependence of Q on temperature were also observed in LVRs using different piezoelectric materials [33,93]. Segovia-Fernandez et al. [31] investigated the effect of temperature on Q in a 1-GHz AlN LVR and attributed the observed temperature-dependence of Q to TED.

Recently, Tu et al. [52] utilized the cryogenic cooling as a practical approach to analyze the dominance of TED in AlN-on-Si LVRs with different physical geometries and vibration modes. Figure 6a,b shows the optical micrograph and the schematic of the rectangular LVR under investigation, respectively. Two resonators were designed with different supporting tether lengths (Design A: La = 10 μm; Design B: La = 45 μm). Figure 6c depicts the symmetrical and anti-symmetrical vibration modes present in both designs. It was experimentally found that the measured values of unloaded quality factor (Qu) at room temperature for both symmetrical and anti-symmetrical vibration modes in Design A were substantially larger than that of Design B. To clarify the dominant dissipation mechanisms in these two designs with only difference in supporting tether length, cryogenic cooling was used as an experimental investigation means. Figure 6d shows the measured electrical transmission magnitudes (S21) for the symmetrical mode as one device of Design A is cooled from room temperature to 78 K. It can be seen that operating the device at 78 K doubles the measured Qu compared to that at room temperature (298 K), and the insertion loss are accordingly reduced by 4 dB. Same cryogenic cooling experiment was repeated over three die samples and consistent results were obtained as shown in Figure 6e, where a linear relationship between Qu and 1/T can be observed. By contrast, Qu remained almost constant for devices of Design B upon the same experiment. This suggests that Qu of Design B is set by anchor loss which is temperature-independent. For Design A,

Figure 5. Evaluation of thermoelastic damping (TED) using numerical analysis: (a) computed thermalprofile showing temperature gradients as a result of local compression and tension; (b) computedthermal profile showing that maximum change in temperature occurs at electrodes; (c) computedQTED for both flat-edge and biconvex resonators as a function of temperature; (d) computed QTED as afunction of thermal expansion coefficient of AlN layer (αAlN) for resonators with different curvatures(κ). This figure is reproduced from [50] (© 2019 IOP Publishing).

The key challenge to experimentally investigate TED in MEMS resonators is that TED is typicallypresent together with anchor loss, which makes it difficult to analyze TED separately. Thus, it isimportant to find an approach to separate these two dissipation mechanisms. It is reported byLi et al. [92] that the measured Q was increased by about 2.5 times when operating a 61-MHzpolysilicon wine-glass disk micro-resonators at cryogenic temperatures down to 5 K compared to roomtemperature. This experimental result suggests that at least one of the dominant dissipation mechanismsin silicon micro-resonators is dependent on temperature. This dependence of Q on temperature werealso observed in LVRs using different piezoelectric materials [33,93]. Segovia-Fernandez et al. [31]investigated the effect of temperature on Q in a 1-GHz AlN LVR and attributed the observedtemperature-dependence of Q to TED.

Recently, Tu et al. [52] utilized the cryogenic cooling as a practical approach to analyze thedominance of TED in AlN-on-Si LVRs with different physical geometries and vibration modes.Figure 6a,b shows the optical micrograph and the schematic of the rectangular LVR under investigation,respectively. Two resonators were designed with different supporting tether lengths (Design A:La = 10 µm; Design B: La = 45 µm). Figure 6c depicts the symmetrical and anti-symmetrical vibrationmodes present in both designs. It was experimentally found that the measured values of unloadedquality factor (Qu) at room temperature for both symmetrical and anti-symmetrical vibration modesin Design A were substantially larger than that of Design B. To clarify the dominant dissipationmechanisms in these two designs with only difference in supporting tether length, cryogenic cooling

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was used as an experimental investigation means. Figure 6d shows the measured electrical transmissionmagnitudes (S21) for the symmetrical mode as one device of Design A is cooled from room temperatureto 78 K. It can be seen that operating the device at 78 K doubles the measured Qu compared to that atroom temperature (298 K), and the insertion loss are accordingly reduced by 4 dB. Same cryogeniccooling experiment was repeated over three die samples and consistent results were obtained asshown in Figure 6e, where a linear relationship between Qu and 1/T can be observed. By contrast,Qu remained almost constant for devices of Design B upon the same experiment. This suggests thatQu of Design B is set by anchor loss which is temperature-independent. For Design A, TED plays amore significant role in setting Qu at room temperature, and TED become smaller as temperature werelowered. These experimental results taken as a whole demonstrate that cryogenic cooling can be usedas an effective approach to study the dominance of respective damping sources in LVRs.

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TED plays a more significant role in setting Qu at room temperature, and TED become smaller as temperature were lowered. These experimental results taken as a whole demonstrate that cryogenic cooling can be used as an effective approach to study the dominance of respective damping sources in LVRs.

Figure 6. Experimental investigation of thermoelastic damping (TED) using cryogenic cooling: (a) optical micrograph of a rectangular Al-on-Si LVR with supporting tethers (length is La) at two sides; (b) schematic of the resonator structure showing constituent layers; (c) simulated symmetrical and anti-symmetrical vibration modes at 47.6 MHz and 46.8 MHz, respectively; (d) measured electrical transmission magnitudes (S21) for the symmetrical mode as the device is cooled from room temperature to 78 K; (e) mean values and standard deviations of measured Q for the resonator with La = 10 μm when cooled from 300 K to 78 K; (f) mean values of measured Q for the resonator with La

= 45 μm when cooled from 300 K to 78 K. This figure is reproduced from [52] (© 2016 Elsevier).

4. Q-enhancement Strategies for LVRs

Over the last decade, numerous Q-enhancement strategies have been proposed to reduce either anchor loss or TED, which are widely considered as the two of the most dominant dissipation mechanisms in LVRs. This section will review some of the Q-enhancement strategies.

4.1. Q-enhancement Strategies for Reducing Anchor Loss

Up to now, most of the reported strategies on reducing anchor loss share the same rationale which is to confine the acoustic energy inside the resonator body and prevent it from leaking to the surrounding substrate. To this end, acoustic reflectors, which serve to reflect acoustic energy, are commonly introduced along the acoustic wave travelling path from the resonator body to the substrate. The acoustic reflectors can be either placed at the substrate area or supporting tethers. An alternative method to confine the acoustic energy inside the resonator body is to modify the geometry of the resonator body directly. Some of the work based on these Q-enhancement strategies are reviewed as follows.

Harrington et al. [94] demonstrated a type of arc-shape acoustic reflectors in AlN-on-Si LVRs. The introduction of etched arcs creates a mismatch in acoustic impedances, which is utilized to reflect the acoustic energy back to the resonator body. It is reported in [94] that the acoustic reflector should be positioned at half a wavelength to the resonator in order to obtain largest Q enhancement. Recently, Hakhamanesh et al. [95] further developed this Q-enhancement strategy in a 86-MHz AlN-on-Si LVR by introducing the etched notches with specific angles, which is believed to provide smooth acoustic impedance transformation and thus enable higher Q-enhancement. Figure 7a depicts the SEM of the device, where it can be seen that the arcs and notches are etched through the substrate. It was shown that the measured Q (19,700) of the LVR was 13.1 times that without arc-shape reflectors and notches (1500) [95]. Instead of etching voids (i.e., subtractive process), the acoustic reflectors can also be

Figure 6. Experimental investigation of thermoelastic damping (TED) using cryogenic cooling:(a) optical micrograph of a rectangular Al-on-Si LVR with supporting tethers (length is La) at two sides;(b) schematic of the resonator structure showing constituent layers; (c) simulated symmetrical andanti-symmetrical vibration modes at 47.6 MHz and 46.8 MHz, respectively; (d) measured electricaltransmission magnitudes (S21) for the symmetrical mode as the device is cooled from room temperatureto 78 K; (e) mean values and standard deviations of measured Q for the resonator with La = 10 µmwhen cooled from 300 K to 78 K; (f) mean values of measured Q for the resonator with La = 45 µmwhen cooled from 300 K to 78 K. This figure is reproduced from [52] (© 2016 Elsevier).

4. Q-Enhancement Strategies for LVRs

Over the last decade, numerous Q-enhancement strategies have been proposed to reduce eitheranchor loss or TED, which are widely considered as the two of the most dominant dissipationmechanisms in LVRs. This section will review some of the Q-enhancement strategies.

4.1. Q-Enhancement Strategies for Reducing Anchor Loss

Up to now, most of the reported strategies on reducing anchor loss share the same rationalewhich is to confine the acoustic energy inside the resonator body and prevent it from leaking tothe surrounding substrate. To this end, acoustic reflectors, which serve to reflect acoustic energy,are commonly introduced along the acoustic wave travelling path from the resonator body to thesubstrate. The acoustic reflectors can be either placed at the substrate area or supporting tethers.An alternative method to confine the acoustic energy inside the resonator body is to modify thegeometry of the resonator body directly. Some of the work based on these Q-enhancement strategiesare reviewed as follows.

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Harrington et al. [94] demonstrated a type of arc-shape acoustic reflectors in AlN-on-Si LVRs.The introduction of etched arcs creates a mismatch in acoustic impedances, which is utilized to reflectthe acoustic energy back to the resonator body. It is reported in [94] that the acoustic reflector shouldbe positioned at half a wavelength to the resonator in order to obtain largest Q enhancement. Recently,Hakhamanesh et al. [95] further developed this Q-enhancement strategy in a 86-MHz AlN-on-Si LVRby introducing the etched notches with specific angles, which is believed to provide smooth acousticimpedance transformation and thus enable higher Q-enhancement. Figure 7a depicts the SEM ofthe device, where it can be seen that the arcs and notches are etched through the substrate. It wasshown that the measured Q (19,700) of the LVR was 13.1 times that without arc-shape reflectors andnotches (1500) [95]. Instead of etching voids (i.e., subtractive process), the acoustic reflectors can alsobe realized by additive layers. Gao et al. [96] demonstrated an acoustic reflector by adding a Pt layerregionally and lithographically setting the release area in a 250-MHz AlN LVR as shown in Figure 7b.The reported Q enhancement of this method is 3.1 times. Another commonly adopted approach to forman acoustic reflector is to use phononic crystals (PnCs) which feature frequency-defined acoustic bandgaps prohibiting propagation of acoustic waves. To serve as an effective acoustic reflector, the band gapof the PnC should be engineered to cover the frequency of the outgoing acoustic wave. Zhu et al. [97]demonstrated this Q-enhancement strategy in a 143-MHz AlN-on-Si LVR, as shown in Figure 7c,where it can be seen that a two-dimensional (2D) PnC array is formed using periodically arrangedair-holes as unit cell. The measured Q of this device is 3620, which is twice that of the one withoutemploying PnC. Apart from air-holes [98], other shapes of PnC unit cell, such as rings [99], crosses [100],fractals [101], snowflakes [102] and solid-disks [103] were also reported. Among them, it is shownby Ardito et al. [103] that solid-disk PnC unit cell outperforms others in that is has larger band gapwidth, which is believed to help confine the acoustic waves. Recently, Siddiqi et al. [104] demonstratedthis concept in a 141-MHz AlN-on-Si LVR as shown in Figure 7e. In this work, the solid-disk PnC ischaracterized by a wide bandgap of 120 MHz around a center frequency of 144.7 MHz. The measuredenhancement in Q is 4.2 times to realize Qs of about 10,000, which is significantly higher than previousworks, as shown in Figure 7d. This greater Q enhancement confirms that wider bandgap of PnCs helpsreduce anchor loss in LVRs. Another concern in designing PnCs is the input/output (I/O) track routing.Since the width of inter-cell link should be as narrow as possible for wider band gap, it becomesimpractical to route the I/O track through these narrow inter-cell links. This is the reason why somebuffering spaces are left between the resonator and 2D PnC array for I/O track routing as shown inFigure 7c,d. However, this buffer space provides a leakage path for acoustic waves. To address thisdeficiency, Siddiqi et al. [105] proposed a hybrid PnC array formed by two types of PnC unit cells.Figure 7f shows the SEM of the resonator with solid-disk and ring shape PnC arrays at two sides.The ring-shape PnC array serves as an acoustic reflector while also providing wider links for ease ofI/O tracking. The enhancement in Q is reported to be 3.7 times which is comparable to that of the PnCdesign shown in Figure 7e. Instead of placing the acoustic reflectors in the substrate area, it is alsofeasible to integrate the reflectors on the supporting tethers. Sorenson et al. [106] demonstrated a Qenhancement of 1.8 times in a 213-MHz AlN-on-Si LVR by placing acoustic reflectors on supportingtethers. The device is shown in Figure 7g, where it can be seen that a ring-shape 1D array of PnC isused to form the acoustic reflectors. In Sorenson’s work, multiple ring-shape 1D arrays were also usedto form the supporting tethers in a 604-MHz AlN-on-Si LVR, which exhibited a 1.6× enhancement in Q.Another example of a 1D PnC is the cross-shape 1D PnC by Lin et al. [107] applied to a 555-MHz AlNLVR as shown in Figure 7h. The reported Q enhancement was 1.5 times. More recently, Rawat et al. [108]proposed novel asymmetric PnC tethers for reduction of anchor loss in a 313-MHz AlN-on-Si LVRas shown in Figure 7i. It was reported that the device with asymmetric PnC on the tethers exhibiteda measured Q about 24 times that of using simple beam tethers (the reference resonators based onsimple beam tethers had an unusually low Q of around 300). Compared to placing acoustic reflectorsin the substrate area, integrating the reflectors directly on tethers favors a more compact size. However,the long and narrow 1D PnC tethers normally make the device vulnerable to mechanical shock [97].

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Figure 7. Q-enhancement strategies by placing acoustic reflectors on substrate or supporting tether: (a) SEM of an 82-MHz Al-on-Si LVR using etched arcs as acoustic reflectors together with etched notches. This figure is reproduced from [95] (© 2018 IEEE); (b) schematic of a 250-MHz AlN LVR using Pt bricks as acoustic reflectors. This figure is reproduced from [96] (© 2019 IEEE); (c) SEM of a 143-MHz Al-on-Si LVR with 2D phononic crystals (PnC) arrays of air holes. This figure is reproduced from [97] (© 2015 IEEE); (d) SEM of an 101-MHz Al-on-Si LVR with 2D PnC arrays of rings. This figure is reproduced from [99] (© 2016 AIP Publishing); (e) SEM of a 141-MHz Al-on-Si LVR with 2D PnC arrays of solid-disks. This figure is reproduced from [104] (© 2018 MDPI); (f) SEM of a 141-MHz Al-on-Si LVR with hybrid 2D PnC arrays of solid-disks and rings. This figure is reproduced from [105] (© 2019 IEEE); (g) SEM of a 213-MHz AlN-on-Si LVR with ring-shape 1D PnC arrays. This figure is reproduced from [106] (© 2011 IEEE); (h) SEM of a 555-MHz AlN LVR with cross-shape 1D PnC array. This figure is reproduced from [107] (© 2014 IEEE); (i) Optical micrograph of a 313-MHz AlN-on-Si LVR with asymmetric PnC on the tethers. This figure is reproduced from [108] (© 2017 IEEE).

Apart from employing acoustic reflectors in the substrate area and supporting tethers, researchers also resort to more compact design strategies which aim to confine the acoustic energy by making modifications directly on the resonator body. Lin et al. [109] proposed a Q enhancement strategy which utilized biconvex edges instead of conventional flat edges. Figure 8a depicts the schematic and the simulated mode shape of such device. It can be seen that from the simulated mode shape that biconvex edges serve to confine the acoustic energy to the central area of the resonator body and thus prevent the acoustic energy from leaking out through the supporting tethers. It is reported that 2.6× enhancement in Q was obtained in a 492-MHz AlN LVR. This Q-enhancement strategy was also demonstrated in a series of AlN-on-Si LVRs with resonant frequencies from 70 MHz to 140 MHz [110,111]. A systematic study on the key design parameters of the biconvex LVRs such as mode order, resonator length, curvature and electrode coverage was performed by Siddiqi et al. [111]. It has been experimentally demonstrated in Siddiqi’s work that the level of Q enhancement is a function of the curvature of biconvex edges, and the anchor loss can be reduced to a point where TED

Figure 7. Q-enhancement strategies by placing acoustic reflectors on substrate or supporting tether:(a) SEM of an 82-MHz Al-on-Si LVR using etched arcs as acoustic reflectors together with etchednotches. This figure is reproduced from [95] (© 2018 IEEE); (b) schematic of a 250-MHz AlN LVRusing Pt bricks as acoustic reflectors. This figure is reproduced from [96] (© 2019 IEEE); (c) SEM of a143-MHz Al-on-Si LVR with 2D phononic crystals (PnC) arrays of air holes. This figure is reproducedfrom [97] (© 2015 IEEE); (d) SEM of an 101-MHz Al-on-Si LVR with 2D PnC arrays of rings. This figureis reproduced from [99] (© 2016 AIP Publishing); (e) SEM of a 141-MHz Al-on-Si LVR with 2D PnCarrays of solid-disks. This figure is reproduced from [104] (© 2018 MDPI); (f) SEM of a 141-MHzAl-on-Si LVR with hybrid 2D PnC arrays of solid-disks and rings. This figure is reproduced from [105](© 2019 IEEE); (g) SEM of a 213-MHz AlN-on-Si LVR with ring-shape 1D PnC arrays. This figure isreproduced from [106] (© 2011 IEEE); (h) SEM of a 555-MHz AlN LVR with cross-shape 1D PnC array.This figure is reproduced from [107] (© 2014 IEEE); (i) Optical micrograph of a 313-MHz AlN-on-SiLVR with asymmetric PnC on the tethers. This figure is reproduced from [108] (© 2017 IEEE).

Apart from employing acoustic reflectors in the substrate area and supporting tethers, researchersalso resort to more compact design strategies which aim to confine the acoustic energy by makingmodifications directly on the resonator body. Lin et al. [109] proposed a Q enhancement strategywhich utilized biconvex edges instead of conventional flat edges. Figure 8a depicts the schematicand the simulated mode shape of such device. It can be seen that from the simulated mode shapethat biconvex edges serve to confine the acoustic energy to the central area of the resonator bodyand thus prevent the acoustic energy from leaking out through the supporting tethers. It is reportedthat 2.6× enhancement in Q was obtained in a 492-MHz AlN LVR. This Q-enhancement strategywas also demonstrated in a series of AlN-on-Si LVRs with resonant frequencies from 70 MHz to140 MHz [110,111]. A systematic study on the key design parameters of the biconvex LVRs such asmode order, resonator length, curvature and electrode coverage was performed by Siddiqi et al. [111].It has been experimentally demonstrated in Siddiqi’s work that the level of Q enhancement is afunction of the curvature of biconvex edges, and the anchor loss can be reduced to a point where

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TED becomes observable. Figure 8b shows one of such LVR which yields 3.8× enhancement in Qcompared to flat-edge device. This energy localization phenomenon was analyzed using dispersiontheory described by Tabrizian et al. [112], who shows that transformation of propagating waves inthe central region to evanescent waves at the flanks occurs when the cross-sectional dimensions ofthe acoustic waveguide changes. This indicates that the physical dimensions of the flanks no longerimpact the intended vibration mode if only evanescent waves are present at the flanks. This idea hasled to an interesting work by Ghatge et al. [113,114], which adopted wide tethers to support LVRsinstead of using narrow tethers as shown in Figure 8c. The benefits of employing wide tethers includeenhancement of power handling capability, less vulnerability to mechanical shock and convenience inrouting the I/O electrodes. Instead of designing gradually-changed width in the acoustic waveguides,Zou et al. [115,116] and Lin et al. [117] demonstrated Q enhancements in AlN LVRs using more abruptvariations in geometry at two ends of the resonator body such as design with beveled, rounded andchamfered corners as shown in Figure 8d–f. Among these designs, the LVR design using roundedcorner exhibited largest Q enhancement of 1.6 times. Some design strategies based on local etchingwere also proposed to reduce anchor loss such as introducing etched slots [118] or etched holes [119]as shown in Figure 8g,h. Apart from the strategies mentioned above, making modification to IDTelectrodes were also proposed as an effective means to suppress spurious resonances and thus enhanceQ [120,121]. Giovannini et al. [120] demonstrated a 1.2× enhancement of Q in a 889-MHz AlN LVRusing apodized electrode configuration as shown in Figure 8i. This work shows that elimination ofspurious resonances helps increase Q by focusing more acoustic energy into the intended resonantpeak. However, the apodized electrode configuration usually suffers from reduction of keff

2. To boost Qwithout degrading keff

2, Zhou et al. [121] recently proposed a spurious resonance suppression techniquefor AlN LVR using IDT electrodes with hammerhead structure at the end. This idea borrows from thedesign of a Piston mode structure, which has been widely used to suppress transverse modes in SAWand FBAR devices [122,123]. Table 1 provides a summary of various Q-enhancement strategies forreducing anchor loss in LVRs.

Table 1. Summary of Q-enhancement strategies for reducing anchor loss in LVRs.

Acoustic Reflector Layered Structure Resonant Frequency Q (Enhanced) Q (Reference) Increase in Q Reference

Arc-shape AlN-on-Si 110 MHz 12042 1818 × 6.6 [94]Arc-shape AlN-on-Si 27 MHz 21000 16000 × 1.3 [94]Arc-shape AlN-on-Si 86 MHz 19700 1500 × 13.1 [95]

Metal brick AlN 250 MHz 3000 980 × 3.1 [96]2D PnC

(air hole) AlN-on-Si 142 MHz 3620 1822 × 2.0 [97]

2D PnC(ring) AlN-on-Si 101 MHz 5369 2012 × 2.7 [99]

2D PnC(solid disk) AlN-on-Si 141 MHz 10492 2510 × 4.2 [104]

2D PnC(disk+ring) AlN-on-Si 141 MHz 10000 2700 × 3.7 [105]

1D PnC(ring) AlN-on-Si 213 MHz 2550 1400 × 1.8 [106]

1D PnC(ring) AlN-on-Si 604 MHz 11400 7200 × 1.6 [106]

1D PnC(cross) AlN 555 MHz 2773 1849 × 1.5 [107]

1D PnC(asymmetric) AlN 313 MHz 3482 313 × 24 [108]

Biconvex edges

AlN 492 MHz 3280 1255 × 2.6 [109]AlN-on-Si 71 MHz 7500 1000 × 7.5 [110]AlN-on-Si 106 MHz 7350 1500 × 4.9 [110]AlN-on-Si 141 MHz 10894 2866 × 3.8 [111]

Varied width AlN-on-Si 82 MHz 18955 NA NA [113]Beveled shape AlN 863 MHz 4189 3348 × 1.3 [115]

Rounded shape AlN 864 MHz 5352 3348 × 1.6 [115]Chemfered corner AlN 829 MHz 3016 2041 × 1.5 [117]

Etched slots AlN 220 MHz 3902 2360 × 1.7 [118]Etched holes AlN-on-Si 105 MHz 8000 1400 × 5.7 [119]

Apodized electrodes AlN 889 MHz 1849 1585 × 1.2 [120]

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becomes observable. Figure 8b shows one of such LVR which yields 3.8× enhancement in Q compared to flat-edge device. This energy localization phenomenon was analyzed using dispersion theory described by Tabrizian et al. [112], who shows that transformation of propagating waves in the central region to evanescent waves at the flanks occurs when the cross-sectional dimensions of the acoustic waveguide changes. This indicates that the physical dimensions of the flanks no longer impact the intended vibration mode if only evanescent waves are present at the flanks. This idea has led to an interesting work by Ghatge et al. [113,114], which adopted wide tethers to support LVRs instead of using narrow tethers as shown in Figure 8c. The benefits of employing wide tethers include enhancement of power handling capability, less vulnerability to mechanical shock and convenience in routing the I/O electrodes. Instead of designing gradually-changed width in the acoustic waveguides, Zou et al. [115,116] and Lin et al. [117] demonstrated Q enhancements in AlN LVRs using more abrupt variations in geometry at two ends of the resonator body such as design with beveled, rounded and chamfered corners as shown in Figure 8d–f. Among these designs, the LVR design using rounded corner exhibited largest Q enhancement of 1.6 times. Some design strategies based on local etching were also proposed to reduce anchor loss such as introducing etched slots [118] or etched holes [119] as shown in Figure 8g,h. Apart from the strategies mentioned above, making modification to IDT electrodes were also proposed as an effective means to suppress spurious resonances and thus enhance Q [120,121]. Giovannini et al. [120] demonstrated a 1.2× enhancement of Q in a 889-MHz AlN LVR using apodized electrode configuration as shown in Figure 8i. This work shows that elimination of spurious resonances helps increase Q by focusing more acoustic energy into the intended resonant peak. However, the apodized electrode configuration usually suffers from reduction of keff2. To boost Q without degrading keff2, Zhou et al. [121] recently proposed a spurious resonance suppression technique for AlN LVR using IDT electrodes with hammerhead structure at the end. This idea borrows from the design of a Piston mode structure, which has been widely used to suppress transverse modes in SAW and FBAR devices [122,123]. Table 1 provides a summary of various Q-enhancement strategies for reducing anchor loss in LVRs.

Figure 8. Q-enhancement strategies by geometry modification on the resonator body: (a) schematic and simulated mode shape of a 492-MHz AlN LVR using biconvex edges. This figure is reproduced Figure 8. Q-enhancement strategies by geometry modification on the resonator body: (a) schematicand simulated mode shape of a 492-MHz AlN LVR using biconvex edges. This figure is reproducedfrom [109] (© 2011 AIP Publishing); (b) SEM of a 142-MHz AlN-on-Si LVR using biconvex edges.This figure is reproduced from [111] (© 2018 IOP Publishing); (c) SEM of a 82-MHz AlN-on-Si LVRusing gradually-changed width. This figure is reproduced from [113] (© 2019 IEEE); (d) SEM ofa 863-MHz AlN LVR with beveled butterfly shape; (e) SEM of a 863-MHz AlN LVR with roundedbutterfly shape. (d,e) are reproduced from [115] (© 2017 IEEE); (f) SEM of a 830-MHz AlN LVR withchamfered corners. This figure is reproduced from [117] (© 2015 IEEE); (g) simulated vibration modeshape of a 220-MHz AlN LVR with etched slot. This figure is reproduced from [118] (© 2013 IEEE);(h) optical micrograph of a 105-MHz AlN-on-Si LVR with etched holes. This figure is reproducedfrom [119] (© 2017 Elsevier); (i) SEM of a 889-MHz AlN LVR using apodized electrode configuration.This figure is reproduced from [120] (© 2014 Elsevier).

4.2. Q-Enhancement Strategies for Reducing TED

While much work has been done to reduce anchor loss, there has been comparatively far lesswork reported on suppressing TED. This is partly due to the fact that the important role of TED insetting Q of LVRs was realized rather recently [50,51,111]. Given the dependence of TED on metalelectrodes, a straightforward strategy to reduce TED is to decrease of electrode coverage in LVRs.Segovia-Fernandez et al. [51] studied a 1-GHz AlN LVR using Au as top electrode and Pt as bottomelectrode, and showed that the measured Qu was doubled when electrode coverage ratio drop from75% to 25%. This Q enhancement by reducing the electrode coverage was consistently observed whenPt and Al were adopted as the top electrode materials. It was also experimentally demonstratedby Siddiqi et al. [111] in a series of biconvex 142-MHz AlN-on-Si LVRs that the measured Qu canbe increased by 60~70% when reducing the number of IDT electrodes from 5 to 3. In comparison,no significant effect in measured Qu can be observed for conventional flat-edge LVRs, which indicatedthat the dependence of TED on electrode coverage was more notable as anchor loss was minimized.

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However, reducing the electrode coverage inevitably increases the motional resistance of LVRs.Apart from reducing the electrode coverage, it is possible to suppress TED by choosing metal electrodematerials with optimal thermal characteristics [51]. However, complexity arises in that there are manyother factors such as resistivity, acoustic impedance and power durability that need to be consideredwhen choosing the electrode materials. In addition, Yan et al. [56] demonstrated a 940-MHz AlNLVR with non-contacting IDT electrodes, which exhibited 5× enhancement in Q compared to theconventional LVR using contacting IDT electrodes. This novel design separates the piezoelectricfilm from its electrodes by sub-micron gaps, which eliminates the mechanical coupling between thepiezoelectric film and the electrodes while keeps the electrical coupling still effective. Although notclarified in [56], it is reasonable to attribute the observed Q enhancement to reduction of TED since theelectrodes no longer displace after physical separation with piezoelectric film. The problem associatedwith this non-contacting electrode design is the complex fabrication process as well as the considerabledrop in the keff

2 due to the air gap.

5. Discussion

5.1. Development for Dissipation Analysis Methods

Although some numerical and experimental analysis methods have been proposed for studyinganchor loss and TED in LVRs as detailed in Section 3, this research field is still in its infant stage.For numerical analysis methods, the validation of computed values of Qanchor and QTED depends highlyon the measured values of Qs that are usually very sensitive to fabrication tolerance, especially forthe case where high-Q resonators are concerned. Given expected device-to-device variations for Qin actual devices, acquiring reliable experimental data to validate models requires at the very leasta pool of measured data from which mean values and variances can be obtained [50,51]. It is notedthat the research community has not yet reached a consensus on conclusive numerical methods forcomputing Qanchor and QTED as much work still needs to be done in further validation of each proposednumerical method on a larger test pool of devices. For experimental analysis methods, cryogeniccooling and out-of-plane displacement profiling methods are effective in clarifying the dominant roleof respective dissipation mechanism (i.e., anchor loss or TED) in LVRs. However, both experimentalanalysis methods currently only yield qualitative results. Further work is required to exploit thesemethods to provide accurate predictions of Qanchor and QTED. In addition, this paper focuses onanalyzing anchor loss and TED, which are two dominant dissipations that set the Q in LVRs operatingin air while having characteristic dimensions at the scale of microns. When operating LVRs in liquidmedium or reducing its critical dimensions to the nano-scale, it can be envisaged that other sources ofdamping such as viscous loss or surface loss would play more dominant roles than anchor loss or TED.As such, numerical and experimental analysis methods targeting other dissipation mechanisms arealso highly relevant in this developing field.

5.2. Comparison of Different Q-Enhancement Strategies

To suppress anchor loss, acoustic reflectors are either placed at the substrate area or realizedas the supporting tethers of LVRs. The former unavoidably increases the footprint of the device.The latter combines the acoustic reflector together with the function of mechanical support achieving aspace-efficient solution. However, building the delicate structures required for the acoustic reflectoron tethers usually compromise the robustness of the mechanical support, making it vulnerable tomechanical shock. Among these strategies based on acoustic reflectors, PnCs have been extensivelyresearched during the past decade [97,100,104,105]. The obvious advantage of using PnCs as acousticreflectors is that it usually offers a frequency stopband that prevents the propagation of elastic wave inany type (i.e., longitudinal or shear wave). In comparison, the arc-shape acoustic reflectors placed athalf-wavelength distance from the resonator is only aimed at reflecting the longitudinal wave in anarrow frequency band, which falls short of PnCs in terms of reflecting efficiency. One problem for

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acoustic reflectors based on PnCs is that their stopband characteristics such as center frequency andbandwidth highly depend on the dimensions of the unit cell structure. Generally, smaller dimensionsare required for higher center frequency or larger bandwidth, which increases the complexity of themicrofabrication process.

An alternative strategy to reduce anchor loss is to modify the resonator body itself. This strategyrelies on the efficient confinement of acoustic energy within the resonator body by designing appropriatemechanical boundaries or electrical excitation configurations. The biconvex designs [109–111] andthe designs with gradual change in width [113,114] all share the same rationale which serves totransform the propagating waves in the central region to evanescent waves at the flanks in order toreduce the leakage of acoustic energy. This strategy has been demonstrated to work across a rangeof frequencies. However, one problem of this strategy is that sufficient area is required for effectivetransformation of propagating waves to evanescent waves, which essentially reduces the maximumpiezoelectric transduction area achievable [111,114]. There are some other design strategies, which arebased on more abrupt geometric modifications, proposed to reduce anchor loss, such as using beveled,rounded or chamfered corners at the ends of the resonator bodies [115–117] and introducing etchedslots or holes [118,119], but the broad applicability of these methods to different operating frequencieshas not been investigated to date. Another strategy to enhance Q involves designing the electricalexcitation configuration to suppress spurious modes which could carry the acoustic energy awayfrom the resonator body [120,121]. This Q-enhancement strategy is attractive for LVRs using moreanisotropic piezoelectric materials as the resonator body. For instance, it has been reported that LVRsbased on single crystal Lithium Niobate (sc-LN) film suffer from vulnerability to spurious modes [124].

5.3. Q-Enhancement Strategies for Novel LVRs

This paper mainly focuses on the work reported for LVRs utilizing polycrystalline AlN thinfilm and operating in the S0 mode of Lamb wave. Among the different Lamb waves, the S0 modein AlN or sc-LN thin films has received the widest attention due to its suitable acoustic propertiessuch as moderate phase velocity and low dispersion. However, attaining large keff

2 at several GHz,which is highly desired for the application of wide-band RF filters, is difficult for S0 mode. Recently,sc-LN LVRs utilizing the first-order asymmetric modes (A1) have gained considerable attention due toits high acoustic velocity (>10,000 m/s) and large electromechanical coupling (>10%) [125,126]. It hasbeen demonstrated that sc-LN LVRs operating in A1 mode can achieve resonant frequency as highas 5 GHz with keff

2 of 25% [127,128]. However, the reported Qs of sc-LN LVRs remain comparativelylow relative to AlN LVRs. Although the dominant damping mechanism in A1-mode sc-LN LVRsremains unclear, it was found the Q can be improved by suppressing the spurious modes, which wasrealized by introducing etched windows between the electrodes and bus-bar, as well as proper designof the electrodes [125]. This Q-enhancement strategy essentially relies on optimizing the mechanicalboundary and electrical excitation configuration.

Another novel type of LVRs that has been heavily researched recently involves the use of Sc-dopedAlN (ScAlN) piezoelectric films aimed at enhancing keff

2. Although it has been reported that thekeff

2 can be increased by about four times compared to un-doped AlN for a doping concentrationof 40%, the improvement comes at the cost of significant reduction in Q [129]. Possible reasons fordegradation in Q are related to the formation of inclusions and film stress during the film depositionprocess [130]. Besides optimizing the process parameters to avoid inclusions, as well as controllingfilm stress, an interesting Q-enhancement strategy in terms of materials advances has been to combinesthe ScAlN film with single crystal silicon to realize high-Q ScAlN-on-Si TPoS LVRs [131].

6. Conclusions

Over the past two decades, substantial progress has been made in the research field of piezoelectricMEMS resonators, especially for LVRs. Piezoelectric LVRs feature wide operation frequency coverage(from tens of MHz to several GHz) and provide a pathway towards realizing a multi-frequency

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integrated solution on a single chip. Significant research effort has been devoted to applying LVRs torealize high-performance resonant sensors, RF filters and oscillators. Among all these applications,LVRs with high Q are always desired, spurring great interest in elucidating and modeling dominantdamping mechanisms in LVRs, based on which appropriate and targeted Q-enhancement strategies canbe formulated and validated. This paper has focused on reviewing the analysis methods for anchor lossand TED, which are known to dominate damping and limit Q in most of LVRs. Various Q-enhancementstrategies targeting reducing these two types of losses have also been reviewed and discussed here.Current outstanding issues and potential research directions are highlighted. In particular, in the caseof LVRs based on sc-LN and ScAlN thin films that have recently garnered much interest, much workstill needs to be done to clarify and understand their underlying dominant sources of damping that setQ. We foresee that effective dissipation analysis methods and Q-enhancement strategies for these novelLVRs will be in high demand given their highly promising applications in 5G RF filters.

Funding: This work was supported by research start-up grant from University of Electronic Science andTechnology of China (No. Y030202059018020), National Natural Science Foundation of China (No. 61804023,No. 61971108), Key R&D Program of Sichuan Province (No. 2018GZ0527, No. 2020ZHCG0038), Sichuan Scienceand Technology Program (No. 2019YJ0198, No. 2020YJ0015), Fundamental Research Funds for the CentralUniversities (No. ZYGX2019Z002) and by a grant by the Research Grants Council of Hong Kong under projectnumber CityU 11204119.

Conflicts of Interest: The authors declare no conflict of interest.

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